Linear form does not vanish on any component I read Shafarevich basic algebraic geometry on page 70 he says for any $X \subseteq \Bbb{P}^N$ can find a linear form $L$ that does not vanish on any component of $X$, I tried to prove this by induction but failed how to prove this?
 A: On each of the finitely many irreducible components $X_1,\cdots, X_r$ of $X$ choose a point $x_i\in X_i$.   
The set $\mathcal L_i\subsetneq (\mathbb P^N)^\ast$  of hyperplanes $H \subset \mathbb P^N$ containing $x_i$ is a closed, strict subspace of the space $ (\mathbb P^N)^\ast$  of hyperplanes of $\mathbb P^N$ (this space  $ (\mathbb P^N)^\ast$ is known as the dual projective space of  $ \mathbb P^N$)
.
The complement $
\mathcal U_i=(\mathbb P^N)^\ast \setminus \mathcal L_i \subset (\mathbb P^N)^\ast$ is thus a non-empty Zariski open  subset of $(\mathbb P^N)^\ast$: it consists of the hyperplanes of $\mathbb P^N$ not containing $x_i$.
Since $(\mathbb P^N)^\ast$ is irreducible (after all it is isomorphic to $\mathbb P^N$ !), the intersection $\mathcal U=\cap_ {i=1}^n \mathcal U_i\subset (\mathbb P^N)^\ast$ is  non-empty too.   
And now we are done: any $H\in \mathcal U$ is a hyperplane of $\mathbb P^N$ containing none of the $x_i$'s and thus a fortiori containing none of the irreducible components $X_i$  of $X$.
If $L=\sum_{i=0}^N a_iz_i$ is a linear form defining $H$ (i.e. $H$ is defined by $L=0$), then that linear form is a solution to your problem.  
Edit: a technical point and the wisdom of Dieudonné 
The dual projective space of  $ \mathbb P^N$ is the space $(\mathbb P^N)^\ast$ of hyperplanes of $ \mathbb P^N$.
A hyperplane $H\subset  \mathbb P^N$ has an equation $a_0z_0+\cdots +a_Nz_N=0$ and the linear form $a_0z_0+\cdots +a_Nz_N$ gives the point $l_H=[a_0:\cdots :a_N]\in (\mathbb P^N)^\ast$.
If now $P=[p_0:\cdots :p_N]\in \mathbb P^N$ is a fixed point in the original projective space $\mathbb P^N$, the set of hyperplanes of $\mathbb P^N$ containing $P$ is the set $\mathcal L_P\subset  (\mathbb P^N)^\ast$ of  those $l_H=[a_0:\cdots :a_N]\in (\mathbb P^N)^\ast$ such that $\sum_i p_ia_i=0$, and all those $l_H$ constitute a hyperplane $\mathcal L_P\subset (\mathbb P^N)^\ast$ in the dual projective space.
The subtle point here is that the $p_i$'s must be considered to be constants and the $a_i$'s to be variables!   
Dieudonné used to emphasize that a characteristic of modern mathematics is that when mathematicians now write $f(x)$ for a function, it may very well happen that $x$ is fixed whereas $f$ is variable: the delta distribution is an egregious example and here we have another example of Dieudonné's profound observation.
